Question

Use bordered determinants to check the following functions for quasiconcavity and quasiconvexity: (a) \(z=-x^{2}-y^{2} \quad(x, y>0)\) (b) \(z=-(x+1)^{2}-(y+2)^{2} \quad(x, y>0)\)

Short answer

Both functions (a) and (b) are quasiconcave as their bordered Hessians are negative.

Step-by-step solution

Understanding Quasiconcavity and Quasiconvexity

Quasiconcavity means that a function's upper level sets are convex, while quasiconvexity means its lower level sets are convex. A practical approach to determine these properties for twice-differentiable functions is by using the bordered Hessian matrix technique.

Define the Bordered Hessian Matrix

To check quasiconcavity or quasiconvexity using bordered determinants, construct the bordered Hessian matrix, which involves the function's first and second derivatives. For a function \(z = f(x, y)\), define the bordered Hessian as:\[ BH = \begin{bmatrix} 0 & f_x & f_y \ f_x & f_{xx} & f_{xy} \ f_y & f_{yx} & f_{yy} \end{bmatrix} \] Where \(f_x, f_y\) are the first partial derivatives, and \(f_{xx}, f_{yy}, f_{xy}(=f_{yx})\) are the second partial derivatives.

Calculate Derivatives for Function (a)

The given function is \(z = -x^2 - y^2\). Calculate the partial derivatives:- First derivatives: \(f_x = -2x, \; f_y = -2y\)- Second derivatives: \(f_{xx} = -2, \; f_{yy} = -2, \; f_{xy}=f_{yx}=0\)

Formulate the Bordered Hessian for Function (a)

For function (a), substitute the derivatives into the bordered Hessian:\[ BH = \begin{bmatrix} 0 & -2x & -2y \ -2x & -2 & 0 \ -2y & 0 & -2 \end{bmatrix} \] Evaluate the minors of the determinant as required for quasiconcavity or quasiconvexity checks.

Evaluate Determinants for Function (a)

Evaluate the determinant of the 2x2 minor:\[ H_2 = \begin{vmatrix} -2 & 0 \ 0 & -2 \end{vmatrix} = 4 \]And the 3x3 bordered determinant:\[ BH = \begin{vmatrix} 0 & -2x & -2y \ -2x & -2 & 0 \ -2y & 0 & -2 \end{vmatrix} = -8 \]Since the main determinant is negative, the function is quasiconcave.

Calculate Derivatives for Function (b)

Given function \(z = -(x+1)^2 - (y+2)^2\). Calculate the partial derivatives:- First derivatives: \(f_x = -2(x+1), \; f_y = -2(y+2)\)- Second derivatives: \(f_{xx} = -2, \; f_{yy} = -2, \; f_{xy}=f_{yx}=0\)

Formulate the Bordered Hessian for Function (b)

For function (b), substitute the derivatives into the bordered Hessian:\[ BH = \begin{bmatrix} 0 & -2(x+1) & -2(y+2) \ -2(x+1) & -2 & 0 \ -2(y+2) & 0 & -2 \end{bmatrix} \] Evaluate the minors of the determinant as required for quasiconcavity or quasiconvexity checks.

Evaluate Determinants for Function (b)

Evaluate the determinant of the 2x2 minor:\[ H_2 = \begin{vmatrix} -2 & 0 \ 0 & -2 \end{vmatrix} = 4 \]And the 3x3 bordered determinant:\[ BH = \begin{vmatrix} 0 & -2(x+1) & -2(y+2) \ -2(x+1) & -2 & 0 \ -2(y+2) & 0 & -2 \end{vmatrix} = -8 \]Since the main determinant is negative, the function is quasiconcave.

Key concepts

Quasiconcavity

Quasiconcavity is a property of some functions where the upper level sets are convex. This means that if you pick any two points on the function's graph, the line segment connecting them will not cross below the graph. This can be visually imagined as a shape that is bowl-like from below. Quasiconcavity is a helpful property in optimization problems because it ensures that local maximizers are also global.

• The upper level sets refer to the set of points where the function is greater than or equal to a given value.
• A convex set implies that for any two points within the set, a straight line can be drawn between them that stays entirely within the set.

The bordered Hessian matrix is often used to test for quasiconcavity in twice-differentiable functions. If the determinant of the bordered Hessian is negative, and its leading principal minors alternate in sign, the function is quasiconcave. This tool simplifies evaluating whether a function has the quasiconcave property.

Quasiconvexity

Quasiconvexity, the sister concept of quasiconcavity, deals with a function's lower level sets being convex. Imagine tracing a line between any two points on a graph, where this line never crosses above the graph itself. This shape is more bowl-like from above. Quasiconvexity is an important concept in economics and optimization as it guarantees that local minimizers are also global.

• The lower level sets are sections of the graph where the function takes on values less than or equal to a specified value.
• Just like with convex sets in quasiconcavity, convexity in these lower sets also implies a straight line between any two points will remain within the set.

Verifying quasiconvexity utilizes similar mathematical tools as quasiconcavity. A bordered Hessian matrix plays a crucial role, where if the pattern of sign changes in principal minors is constant rather than alternating, the function is deemed quasiconvex. Ensuring the right sequence of determinant signs is key in this assessment.

Partial Derivatives

Partial derivatives represent how a multivariable function changes as one of the variables changes, holding the others constant. They are vital in identifying slopes and optimal points for functions involving multiple variables.

• In mathematical notation, the partial derivative of a function with respect to a variable is often denoted by symbols like \( f_x \) or \( \frac{\partial f}{\partial x} \).
• First derivatives provide the slope, or rate of change, while second derivatives offer insights into the function's curvature.

In the bordered Hessian matrix used for quasiconcavity and quasiconvexity tests, both first and second partial derivatives of the function are inserted. Understanding these derivatives allows us to shape the bordered Hessian correctly and explore the critical nature of changes along different paths on a graph.

Determinants

Determinants are mathematical values calculated from square matrices that provide critical insights into geometric and algebraic properties. They are essential in linear algebra for determining matrix invertibility and solutions to system equations.

• The determinant of a matrix is a scalar property that can indicate whether a matrix is non-singular, meaning it has an inverse.
• In the context of Hessian matrices, the sign of the determinant helps establish the concavity properties of functions.

Within the bordered Hessian, determinants of different minor matrices are calculated to test for quasiconcavity and quasiconvexity. Specifically, the size and sign of the determinant assist in deciding whether the function's upper or lower level sets exhibit necessary convexity traits. Evaluating these determinants correctly is fundamental in verifying the function's character.